Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $U$ be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$, and let $f:U\to\mathbb{R}^n$ be a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth} map. Let $a\in U$, and suppose that the \reftext{def:jacobian-determinant-euclidean-open-set-2026a}{Jacobian determinant} satisfies $\det J_f(a)\ne 0$. Then there exist \reftext{def:open-subset-euclidean-space-2026a}{open} sets $V,W\subseteq\mathbb{R}^n$ with $a\in V\subseteq U$ and $f(a)\in W$ such that $f(V)=W$, the restriction $f|_V:V\to W$ is \reftext{def:bijection-sets-2026a}{bijective}, and its inverse $h=(f|_V)^{-1}:W\to V$ is \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth}. Moreover, the \reftext{def:differentiable-map-at-point-euclidean-2026a}{Jacobian matrix} of $h$ satisfies $J_h(y)=\bigl(J_f(h(y))\bigr)^{-1}$ for all $y\in W$, where $(\cdot)^{-1}$ denotes the \reftext{def:inverse-matrix-invertible-real-square-matrix-2026a}{matrix inverse}.